Open Orbits and Augmentations of Dynkin Diagrams

TL;DR

This paper explores the conditions under which a complex reductive Lie group representation admits an open orbit, linking it to Dynkin diagram augmentations and potential applications in geometric structures.

Contribution

It establishes a precise criterion connecting finite orbit properties of representations with Dynkin diagram augmentations and the existence of larger semi-simple groups.

Findings

V admits an open G_0-orbit if and only if it has finitely many G_0-orbits.

Such representations correspond to augmentations of the Dynkin diagram.

The results have implications for the geometry of manifolds with stable forms.

Abstract

Given any representation V of a complex linear reductive Lie group G_0, we show that a larger semi-simple Lie group G with g=g_0 + V + V* + ..., exists precisely when V has a finite number of G_0-orbits. In particular, V admits an open G_0-orbit. Furthermore, this corresponds to an augmentation of the Dynkin diagram of g_0. The representation theory of g should be useful in describing the geometry of manifolds with stable forms as studied by Hitchin.